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P Buckling P P PCR Alternate paths d As P increases from 0, linear response At PCR alternate solutions are possible Buckling occur Slowly/Rapidly Axial Response Bending Response Beam bending Equation d y M 2 dx EI 2 M y Axial Load causes moment if beam is bent M P Py d y P y 0 2 dx EI Solution: Harmonic Equation 2 y x y P L y A sin P P x B cos x EI EI Solution A=0 Axial Motion only No Buckling Trivial Solution Nastran Solution 101 Using Boundary Conditions y x 0 0 B 0 y x L 0 A sin P L EI Alternate Solution - Eigenvalues sin x 2 y A sin P x EI P L n EI Values of P that satisfy above equation are PCR y n 2 2 EI PCR L2 n 1, 2,3... x y P L Typical NASTRAN Deck SOL 105 SPC=# SUBCASE 1 LOAD=## . SUBCASE 2 METHOD=99 . BEGIN BULK EIGRL,99,0.,,3 ENDDATA Constraint Set for model Static Solution (Like Solution 101) Applied Load PAPP Buckling Solution (Eigenvalue Solution) Calls EIGRL card Defines Buckling solution Requests 3 eigen values (buckling modes) Forces l > 0 Output: Subcase 1: Stress due to PAPP Subcase 2: l = eigenvalue (PCR = l PAPP) Linear System K u P K constant u P P Apply Load = PAPP creates PEAK = Peak Stress Apply Load = PULT creates ULT = Ultimate Stress ULT ULT P P ULT PEAK ULT PAPP PEAK PAPP Load that will cause the structure to fail in tension or compression For Buckling P l PAPP CR Load that will cause the structure to fail in buckling Strength of the Structure P*=min(PCR,PULT) Strength-to-Weight Ratio P* W From Mass matrix in f06 file P ULT ULT P PEAK APP Strength-to-Weight Ratio P* W P l PAPP CR P*=min(PCR,PULT) NB: 1. Problem is simplified if PAPP = 1 2. Use Weight density on MAT1 Card (Although Nastran indicates Mass, it is weight) This shortcut is not appropriate for dynamics problems 3. Create a spreadsheet or equivalent tocompute critical parameters and to keep track of designs. Design Approach 1. If PULT < PCR Stress is Critical 1. Increase size of members with high stress Decrease size of members with low stress 2. Add new members to create additional load path 2. If PCR < PULT Buckling is Critical Plot 1st mode shape 1. Add braces to members that buckle Remove braces from members where not needed 2. Increase size of members that buckle Decrease the size of members that don t buckle NB: For Buckling I s 4 s = side dimension of the member PCR 2 2 L = length L L brace at midpoint to maximize effect L L , PCR 4 PCR 2 For Stress F 1 for axial loads A s2 Mc 1 P 3 for bending loads ULT I s Approximate for indeterminate structures P ULT
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Rochester >> ME >> 406 (Fall, 2009)
ME 406 Bifurcations V Subcritical Pitchfork Bifurcation sysid Mathematica 6.0.3, DynPac 11.01, 1122009 intreset; plotreset; imsize = 250; 1. Introduction In this notebook, the fifth in a series of notebooks on bifurcations, we look at a simple exa...
Rochester >> ME >> 6 (Fall, 2009)
ME 406 Bifurcations V Subcritical Pitchfork Bifurcation sysid Mathematica 6.0.3, DynPac 11.01, 1122009 intreset; plotreset; imsize = 250; 1. Introduction In this notebook, the fifth in a series of notebooks on bifurcations, we look at a simple exa...
Rochester >> ME >> 406 (Fall, 2009)
ME 406 Bifurcations IV Supercritical Pitchfork Bifurcation sysid Mathematica 4.1.2, DynPac 10.66, 352002 intreset; plotreset; 1. Introduction In this notebook, the fourth in a series of notebooks on bifurcations, we look at a simple example of a s...
Rochester >> ME >> 5 (Fall, 2009)
ME 406 Bifurcations IV Supercritical Pitchfork Bifurcation sysid Mathematica 4.1.2, DynPac 10.66, 352002 intreset; plotreset; 1. Introduction In this notebook, the fourth in a series of notebooks on bifurcations, we look at a simple example of a s...
Rochester >> ME >> 104 (Fall, 2009)
Most lucrative college degrees \'04 - Feb. 5, 2004 Page 1 of 3 j k l m n CNN/Money n Web i j k l m Autos Real Estate Money\'s Best Home Markets Economy World Biz Technology Commentary Personal Finance College Credit and Debt Insu...
Rochester >> ME >> 241 (Fall, 2009)
On the Role of the Liquid in a Glass Harmonica Roger F. Gans Department of Mechanical Engineering University of Rochester Rochester, NY 14627-0132 March 2005 Abstract The general impression in the literature is that the amount of liquid in the indiv...
Rochester >> ME >> 213 (Fall, 2009)
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Rochester >> ME >> 241 (Fall, 2009)
The Influence of Geometry on Free Convection in Roman Hypocaust Systems Group 3 Mike Brown Eric Konicki Laura Wong Abstract The natural convective flow of the Roman hypocaust system has been subjected to many qualitative studies but few quantitativ...
Rochester >> ME >> 241 (Fall, 2009)
ME 241 Hour Exam Solution Set January 27, 2000 1. What is the campus emergency phone number? 2. Where are the MSDSs for our lab kept? 13 Department Office His/her own safety 3. What is the first safety consideration for any experimenter? For the da...
Rochester >> AST >> 241 (Fall, 2009)
Astronomy 241, Spring 2005 3 March 2005 Today in Astronomy 241: stellar evolution Todays reading: Carroll and Ostlie pp. 483-494, on Evolution on the main sequence Formation of an isothermal, partly degenerate core Hydrogen mass fraction, and H-R ...
Rochester >> AST >> 102 (Fall, 2009)
Today in Astronomy 102: black holes and how to prevent them The Schwarzschild singularity and the sizes of black holes. Degeneracy pressure: a quantum-mechanical effect that might stop matter from collapsing to form a black hole, when gas pressure or...
Rochester >> AST >> 111 (Fall, 2009)
Today in Astronomy 111: Venus and Earth Hot cores of planets: heat from formation, further radioactive heating. Venus and the greenhouse effect Hadley cells: the simple atmospheric circulation patterns in the Venusian atmosphere Earth as a planet Dis...
Rochester >> AST >> 203 (Fall, 2009)
Astronomy 203 practice final examination Fall 1999 If this were a real, in-class examination, you would be reminded here of the exam rules, which are as follows: \"You may consult only one page of formulas and constants, and a calculator, while taking...
Rochester >> AST >> 203 (Fall, 2009)
Astronomy 203/203W practice final exam solutions Fall 1999 No fair peeking at these solutions before attempting to solve the problems yourself! _ 1 (50 points) A 20 cm diameter, 100 cm focal length lens is followed by two smaller lenses, with focal l...
Rochester >> AST >> 111 (Fall, 2009)
Astronomy 111 observing project: solutions Fall 2008 Dan took the data, observed and reduced by the class, from the Project web page, and analyzed them using statistical methods. Statistics is beyond the scope of AST 111, of course, but it will be in...
Rochester >> AST >> 111 (Fall, 2009)
Astronomy 111 Project Manual Fall 2008 Revolution of the moons of Jupiter, Keplers third law, and Newtons law of gravity 1 Introduction This is an observing experiment in which we will determine the mass of Jupiter, by measuring the orbital proper...
Rochester >> AST >> 102 (Fall, 2009)
Today in Astronomy 102: relativity and the Universe General relativity and the Universe. Hubble: the Universe is observed to be homogeneous, isotropic, and expanding Redshift and distance: Hubbles Law Visible-light spectrum of the quasar 3C 273 (Maar...
Rochester >> AST >> 111 (Fall, 2009)
Astronomy 111 Midterm Exam #1: Solutions Fall 2008 If any of these answers seems obscure, please ask questions until we clear it up for you. Recall that the score for this test is based on the best 140 points worth out of the 160 points presented. On...
Rochester >> AST >> 102 (Fall, 2009)
Astronomy 102, Spring 2008 25 March 2008 Today in Astronomy 102: supermassive black holes in active galaxy nuclei (AGNs) Active galaxies: quasars, radio galaxies and their relatives. Why the observations imply that they have supermassive black hole...
Rochester >> PHY >> 123 (Fall, 2009)
Elementary particles Spring 2005, Physics 123 04/10/09 Lecture XXV 1 Concepts antimatter leptons quarks fundamental interactions 04/10/09 Lecture XXV 2 Mass and energy Mass and energy are interchangeable Energy can be used to create mas...
Rochester >> P >> 142 (Fall, 2009)
Workshop module 5 - Physics 142, Fall 2007 1. Batteries are always labeled with their emf; for instance, an AA flashlight battery is labeled \"1.5 volts.\" Would it also be appropriate to put a label on batteries stating how much current they provide?...
Rochester >> P >> 142 (Fall, 2009)
Workshop module 5 - Physics 142, Fall 2008 1. Batteries are always labeled with their emf; for instance, an AA flashlight battery is labeled \"1.5 volts.\" Would it also be appropriate to put a label on batteries stating how much current they provide? ...
Rochester >> PHY >> 114 (Fall, 2009)
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Rochester >> PHY >> 114 (Fall, 2009)
PHY 114 Summer 2005 Lecture Notes Chapter 6 Chapter 6: Magnetic Fields It turns out the hardest thing to understand about magnetism is a simple magnet. We will start by studying the force on a current caused by a magnet field. We\'ll wait until next...
Rochester >> PHY >> 114 (Fall, 2009)
PHY 114 Summer 2005 Lecture Notes Chapter 2 Chapter 2: Gauss Law Lecture Outline 1. 2. 3. 4. The Definition of Electric Flux Gauss Law The Behavior of Conductors Examples Using Gauss Law 1. The Definition of Electric Flux Recall that the strength ...
Rochester >> PHY >> 114 (Fall, 2009)
PHY 114 Summer 2005 Lecture Notes Chapter 3 Chapter 3: Electric Potential The Law of Conservation of Energy is the single most useful law of physics. In this chapter we will develop the insights needed to make use of energy conservation to understa...
Rochester >> PHY >> 114 (Fall, 2009)
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Rochester >> PHY >> 114 (Fall, 2009)
PHY 114 Summer 2005 Syllabus PHY 114 General Physics 2 Outline and Syllabus 1. Contact Information Instructor: Eric Page Office: B&L 463 Phone: 275-8577 Email: epage@pas.rochester.edu IM: ericpage74 on AIM and MSN epage74 on Yahoo! A little about m...
Rochester >> PHY >> 114 (Fall, 2009)
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Rochester >> PHY >> 114 (Fall, 2009)
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Rochester >> PHY >> 114 (Fall, 2009)
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Rochester >> PHY >> 114 (Fall, 2009)
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Rochester >> PHY >> 114 (Fall, 2009)
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Rochester >> PHY >> 114 (Fall, 2009)
PHY 114 Summer 2005 Lecture Notes Chapter 8 Chapter 8: Induction and Inductance Lecture Outline 1. Faraday\'s Law of Induction 2. Motors and Generators 3. The Meissner Effect and Superconductivity 4. The Definition of Mutual Inductance 5. The Defini...
Rochester >> PHY >> 114 (Fall, 2009)
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Rochester >> PHY >> 114 (Fall, 2009)
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Rochester >> PHY >> 114 (Fall, 2009)
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Rochester >> PHY >> 114 (Fall, 2009)
Name: Laboratory Section: Laboratory Section Date: Last Revised on July 6, 2005 Partners Names: Grade: EXPERIMENT 8 Electron Beams 0. Pre-Laboratory Work [2 pts] 1. Why is it necessary to make multiple measurements of the Helmholtz coil diameter? W...
Rochester >> PHY >> 114 (Fall, 2009)
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Rochester >> PHY >> 114 (Fall, 2009)
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Rochester >> PHY >> 114 (Fall, 2009)
PHY 114 Summer 2005 Homework #2 Solutions Required Problems: 1. We find the potential difference from the work-energy principle: Wab = K + U = K + q(Vb Va) 8.00 104 J = 2.10 104 J + ( 8.10 106 C)(Vb Va), Va Vb = + 72.8 V. which gives Vb Va = ...
Rochester >> PHY >> 114 (Fall, 2009)
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Rochester >> PHY >> 114 (Fall, 2009)
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Rochester >> PHY >> 114 (Fall, 2009)
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Rochester >> PHY >> 114 (Fall, 2009)
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Rochester >> PHY >> 114 (Fall, 2009)
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Rochester >> PHY >> 114 (Fall, 2009)
Name:_ PHY 114 Summer 2005 Homework #3 Please hand this in by Tues, July 19th in class. I realize this may seem pretty long, but many of the problems are quite short, and I need to make sure that we cover everything that needs to be covered. Please,...
Rochester >> PHY >> 114 (Fall, 2009)
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Rochester >> CM >> 2007 (Fall, 2009)
Note Set 5 Constrained Optimization 5.1 Overview In this note set, we consider constrained optimization. 5.2 Kuhn-Tucker Conditions Suppose we have an optimization problem of the form, max f ( x) such that ceq ( x) = 0 , cneq ( x) 0 x This ...
Rochester >> SRM >> 2009 (Fall, 2009)
Survey Research Methods PSC 203 Spring 2009 Note Set 2 Descriptive Statistics Survey Research Methods Spring 2009 Note Set 1 Page 2 Outline of Lecture Random variables Graphical description of data Numerical description of data A look at econ...
Rochester >> ADA >> 2007 (Fall, 2009)
Applied Data Analysis PSC 200 Fall 2007 Note Set 9 Inference: Differences in Proportions Applied Data Analysis Fall 2007 Note Set 9 Page 2 Differences in Proportions Example: A pharmaceutical company has developed a new medication that it believ...
Rochester >> ADA >> 2007 (Fall, 2009)
Applied Data Analysis PSC 200 Fall 2007 Note Set 8 Inference: Differences in Means Applied Data Analysis Fall 2007 Note Set 8 Page 2 Overview So far, we have covered two types of hypothesis tests - Tests of a population mean - Tests of a populat...
Rochester >> ADA >> 2007 (Fall, 2009)
Applied Data Analysis PSC 200 Fall 2007 Note Set 10 Inference: Additional Topics Applied Data Analysis Fall 2007 Note Set 11 Page 2 Overview Review of Inference Applications of Hypothesis Testing Small Sample Inference Interpreting Reported R...
Rochester >> ADA >> 2007 (Fall, 2009)
Applied Data Analysis PSC 200 Fall 2007 Note Set 2 Descriptive Statistics Applied Data Analysis Fall 2007 Note Set 2 Page 2 Outline of Lecture Sampling Random variables Graphical description of data Numerical description of data Describe som...
Rochester >> ADA >> 2007 (Fall, 2009)
Applied Data Analysis PSC 200 Fall 2007 Note Set 7 Inference: Proportions Applied Data Analysis Fall 2007 Note Set 7 Page 2 Overview Review proportions Hypothesis testing with proportions Confidence intervals with proportions Applied Data Ana...
Rochester >> ADA >> 2007 (Fall, 2009)
Applied Data Analysis PSC 200 Fall 2007 Note Set 12 Correlation and Bivariate Regression Applied Data Analysis Fall 2007 Note Set 12 Page 2 Overview In this note set, we consider relationships between two variables We discuss: - Hypothesis test...
Rochester >> ADA >> 2006 (Fall, 2009)
Applied Data Analysis Fall 2006 Note Set 17 Page 1 Applied Data Analysis PSC 200 Fall 2006 Note Set 17 Indicator Variables and Analysis of Variance Applied Data Analysis Fall 2006 Note Set 17 Page 2 Qualitative Independent Variables This note ...
Rochester >> ADA >> 2007 (Fall, 2009)
Applied Data Analysis PSC 200 Fall 2007 Note Set 3 Discrete Probability Distributions Applied Data Analysis Fall 2007 Note Set 3 Page 2 Outline of Lecture Review of Sampling Discrete Random Variables Expected Values Some Discrete Distribution...
Rochester >> ADA >> 2006 (Fall, 2009)
Applied Data Analysis PSC 200 Fall 2006 Note Set 12 Surveys Applied Data Analysis Fall 2006 Note Set 12 Page 2 Overview Surveys are one of the most important sources of data for the social sciences Public opinion polls National Election Studie...
Rochester >> ADA >> 2006 (Fall, 2009)
Applied Data Analysis PSC 200 Fall 2006 Note Set 11 Inference: Advanced Topics Applied Data Analysis Fall 2006 Note Set 11 Page 2 Overview Jacobson article Applications of hypothesis testing Sample size determination Size and power of a test ...
Rochester >> ADA >> 2006 (Fall, 2009)
Applied Data Analysis PSC 200 Fall 2006 Note Set 5 Joint Distributions Applied Data Analysis Fall 2006 Note Set 5 Page 2 Outline of Lecture Joint Distributions Independence Measures of Association Applied Data Analysis Fall 2006 Note Set 5 P...
Rochester >> CM >> 2007 (Fall, 2009)
Note Set 4 Unconstrained Optimization 4.1 Overview In this note set, we will consider finding the (unconstrained) maximum of the function f ( x) . The methods we will use will depend on the smoothness of the function. 4.2 Newtons Method The fi...
Rochester >> ADA >> 2006 (Fall, 2009)
Applied Data Analysis PSC 200 Fall 2006 Note Set 6 Sampling Distributions Applied Data Analysis Fall 2006 Note Set 6 Page 2 Outline of Lecture Point Estimators Law of Large Numbers Sampling Distributions The Central Limit Theorem Applied Dat...
Rochester >> ADA >> 2006 (Fall, 2009)
Applied Data Analysis PSC 200 Fall 2006 Note Set 18 Functional Form and Interactions Applied Data Analysis Fall 2006 Note Set 18 Page 2 Nonlinearity In X and Y are related in a linear way, the change in Y resulting form a one unit change in X do...
Rochester >> ADA >> 2006 (Fall, 2009)
Applied Data Analysis PSC 200 Fall 2006 Note Set R2 Review for Second Exam Applied Data Analysis Fall 2006 Note Set R2 Page 2 Exam Overview Exam: - Not cumulative - Open book / open notes / no laptops - Dont practice SPSS for exam - Bring calcul...
Rochester >> ADA >> 2007 (Fall, 2009)
Applied Data Analysis PSC 200 Fall 2007 Note Set 1 Course Overview Applied Data Analysis Fall 2007 Note Set 1 Page 2 Course Overview Instructor: Prof. Michael Peress - OH: Harkness 326, W 3-5 TA: Navine Murshid - OH: Harkness 304, ? Applied Da...
Rochester >> ADA >> 2007 (Fall, 2009)
Applied Data Analysis PSC 200 Fall 2007 Note Set 5 Estimation Applied Data Analysis Fall 2007 Note Set 5 Page 2 Outline of Lecture Measures of Association Point Estimators Law of Large Numbers Sampling Distributions and the Central Limit Theo...
Rochester >> ADA >> 2007 (Fall, 2009)
Applied Data Analysis PSC 200 Fall 2007 Note Set 16 Functional Form and Interactions Applied Data Analysis Fall 2007 Note Set 16 Page 2 Nonlinearity If X and Y are related in a linear way, the change in Y resulting form a one unit change in X do...
Rochester >> ADA >> 2007 (Fall, 2009)
Applied Data Analysis PSC 200 Fall 2007 Note Set 6 Inference: Means Applied Data Analysis Fall 2007 Note Set 6 Page 2 Outline of Lecture Overview of Hypothesis Testing Method 1: P-Values Method 2: Critical Values Method 3: Confidence Interval...
Rochester >> MATH >> 531 (Fall, 2009)
Math 531 Tom Tucker NOTES FROM CLASS 9/22 Note about the homework The following is an immediate consequence of Problems 5 and 6 from the homework you just handed in. Theorem 9.1. Let R be a Noetherian integral domain of dimension 1. Then the followin...
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